Theorem sqabs 11312

Description: The squares of two reals are equal iff their absolute values are equal. (Contributed by NM, 6-Mar-2009.)

Assertion

Ref	Expression
sqabs	|- ( ( A e. RR /\ B e. RR ) -> ( ( A ^ 2 ) = ( B ^ 2 ) <-> ( abs ` A ) = ( abs ` B ) ) )

Proof of Theorem sqabs

Step	Hyp	Ref	Expression
1		resqcl 10733	. . . . 5 |- ( A e. RR -> ( A ^ 2 ) e. RR )
2		sqge0 10742	. . . . 5 |- ( A e. RR -> 0 <_ ( A ^ 2 ) )
3		absid 11301	. . . . 5 |- ( ( ( A ^ 2 ) e. RR /\ 0 <_ ( A ^ 2 ) ) -> ( abs ` ( A ^ 2 ) ) = ( A ^ 2 ) )
4	1, 2, 3	syl2anc 411	. . . 4 |- ( A e. RR -> ( abs ` ( A ^ 2 ) ) = ( A ^ 2 ) )
5		recn 8040	. . . . 5 |- ( A e. RR -> A e. CC )
6		2nn0 9294	. . . . 5 |- 2 e. NN0
7		absexp 11309	. . . . 5 |- ( ( A e. CC /\ 2 e. NN0 ) -> ( abs ` ( A ^ 2 ) ) = ( ( abs ` A ) ^ 2 ) )
8	5, 6, 7	sylancl 413	. . . 4 |- ( A e. RR -> ( abs ` ( A ^ 2 ) ) = ( ( abs ` A ) ^ 2 ) )
9	4, 8	eqtr3d 2239	. . 3 |- ( A e. RR -> ( A ^ 2 ) = ( ( abs ` A ) ^ 2 ) )
10		resqcl 10733	. . . . 5 |- ( B e. RR -> ( B ^ 2 ) e. RR )
11		sqge0 10742	. . . . 5 |- ( B e. RR -> 0 <_ ( B ^ 2 ) )
12		absid 11301	. . . . 5 |- ( ( ( B ^ 2 ) e. RR /\ 0 <_ ( B ^ 2 ) ) -> ( abs ` ( B ^ 2 ) ) = ( B ^ 2 ) )
13	10, 11, 12	syl2anc 411	. . . 4 |- ( B e. RR -> ( abs ` ( B ^ 2 ) ) = ( B ^ 2 ) )
14		recn 8040	. . . . 5 |- ( B e. RR -> B e. CC )
15		absexp 11309	. . . . 5 |- ( ( B e. CC /\ 2 e. NN0 ) -> ( abs ` ( B ^ 2 ) ) = ( ( abs ` B ) ^ 2 ) )
16	14, 6, 15	sylancl 413	. . . 4 |- ( B e. RR -> ( abs ` ( B ^ 2 ) ) = ( ( abs ` B ) ^ 2 ) )
17	13, 16	eqtr3d 2239	. . 3 |- ( B e. RR -> ( B ^ 2 ) = ( ( abs ` B ) ^ 2 ) )
18	9, 17	eqeqan12d 2220	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( A ^ 2 ) = ( B ^ 2 ) <-> ( ( abs ` A ) ^ 2 ) = ( ( abs ` B ) ^ 2 ) ) )
19		abscl 11281	. . . . 5 |- ( A e. CC -> ( abs ` A ) e. RR )
20		absge0 11290	. . . . 5 |- ( A e. CC -> 0 <_ ( abs ` A ) )
21	19, 20	jca 306	. . . 4 |- ( A e. CC -> ( ( abs ` A ) e. RR /\ 0 <_ ( abs ` A ) ) )
22		abscl 11281	. . . . 5 |- ( B e. CC -> ( abs ` B ) e. RR )
23		absge0 11290	. . . . 5 |- ( B e. CC -> 0 <_ ( abs ` B ) )
24	22, 23	jca 306	. . . 4 |- ( B e. CC -> ( ( abs ` B ) e. RR /\ 0 <_ ( abs ` B ) ) )
25		sq11 10738	. . . 4 |- ( ( ( ( abs ` A ) e. RR /\ 0 <_ ( abs ` A ) ) /\ ( ( abs ` B ) e. RR /\ 0 <_ ( abs ` B ) ) ) -> ( ( ( abs ` A ) ^ 2 ) = ( ( abs ` B ) ^ 2 ) <-> ( abs ` A ) = ( abs ` B ) ) )
26	21, 24, 25	syl2an 289	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( abs ` A ) ^ 2 ) = ( ( abs ` B ) ^ 2 ) <-> ( abs ` A ) = ( abs ` B ) ) )
27	5, 14, 26	syl2an 289	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( abs ` A ) ^ 2 ) = ( ( abs ` B ) ^ 2 ) <-> ( abs ` A ) = ( abs ` B ) ) )
28	18, 27	bitrd 188	1 |- ( ( A e. RR /\ B e. RR ) -> ( ( A ^ 2 ) = ( B ^ 2 ) <-> ( abs ` A ) = ( abs ` B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1372   e. wcel 2175   class class class wbr 4043   ` cfv 5268  (class class class)co 5934   CCcc 7905   RRcr 7906   0cc0 7907   <_ cle 8090   2c2 9069   NN0cn0 9277   ^cexp 10664   abscabs 11227

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4478  ax-setind 4583  ax-iinf 4634  ax-cnex 7998  ax-resscn 7999  ax-1cn 8000  ax-1re 8001  ax-icn 8002  ax-addcl 8003  ax-addrcl 8004  ax-mulcl 8005  ax-mulrcl 8006  ax-addcom 8007  ax-mulcom 8008  ax-addass 8009  ax-mulass 8010  ax-distr 8011  ax-i2m1 8012  ax-0lt1 8013  ax-1rid 8014  ax-0id 8015  ax-rnegex 8016  ax-precex 8017  ax-cnre 8018  ax-pre-ltirr 8019  ax-pre-ltwlin 8020  ax-pre-lttrn 8021  ax-pre-apti 8022  ax-pre-ltadd 8023  ax-pre-mulgt0 8024  ax-pre-mulext 8025  ax-arch 8026  ax-caucvg 8027

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4338  df-po 4341  df-iso 4342  df-iord 4411  df-on 4413  df-ilim 4414  df-suc 4416  df-iom 4637  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-rn 4684  df-res 4685  df-ima 4686  df-iota 5229  df-fun 5270  df-fn 5271  df-f 5272  df-f1 5273  df-fo 5274  df-f1o 5275  df-fv 5276  df-riota 5889  df-ov 5937  df-oprab 5938  df-mpo 5939  df-1st 6216  df-2nd 6217  df-recs 6381  df-frec 6467  df-pnf 8091  df-mnf 8092  df-xr 8093  df-ltxr 8094  df-le 8095  df-sub 8227  df-neg 8228  df-reap 8630  df-ap 8637  df-div 8728  df-inn 9019  df-2 9077  df-3 9078  df-4 9079  df-n0 9278  df-z 9355  df-uz 9631  df-rp 9758  df-seqfrec 10574  df-exp 10665  df-cj 11072  df-re 11073  df-im 11074  df-rsqrt 11228  df-abs 11229

This theorem is referenced by:  coskpi  15238